Since reaction is occuring in basic medium, therefore, 2OH− are added on right side. 2MnO4−(aq)+Br−(aq)⟶2MnO2(s)+BrO3−(aq)+2OH−(aq)
Now, hydrogen atoms can be balanced by adding one H2O molecule to the left side 2MnO4−(aq)+Br−(aq)+H2O(l)⟶ 2MnO2(s)+BrO3−(aq)+2OH−(aq)